The Genus Fields of Algebraic Number Fields

abelian extension abelian number field absolute abelian number absolute abelian subfield algebraic number field Chapter class field theory class number composite Corollary cubic number field cyclic over Q cyclotomic number field defining modulus degree q denote Edité Edited Eisenstein type field of degree finite degree finite prime ideals Fröhlich Galois group genus field genus group genus number Hence ideal class group ideal group implies integer Ishida k₁ k₂ Lemma M.Ishida Math minimal polynomial mod q number q Number Theory odd prime degree odd prime number P₁ pe(p prime divisor primitive element Proceedings 1974 Proof q is totally quadratic number field ramification index ramified in K rational prime number satisfied Séminaire Theorem totally ramified type with respect unique subfield unrami unramified abelian extension VIII вк ск